Decision-Making Approach with Fuzzy Type-2 Soft Graphs

Molodtsov’s theory of soft sets is free from the parameterizations insufficiency of fuzzy set theory. Type-2 soft set as an extension of a soft set has an essential mathematical structure to deal with parametrizations and their primary relationship. Fuzzy type-2 soft models play a key role to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. In this research article, we introduce the concept of fuzzy type-2 soft set by integrating fuzzy set theory and type-2 soft set theory. We also introduce the notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We construct some operations such as union, intersection, AND, and OR on fuzzy type-2 soft graphs and discuss these concepts with numerical examples. )e fuzzy type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors structure and is applicable in computational analysis, applied intelligence, and decision-making problems. We study the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services.


Introduction
Fuzzy set theory has its remarkable origin to the work of Zadeh [1] in 1965 to interact with vagueness and imprecision between absolute true and absolute false. e range of the values in a fuzzy set lies in [0,1]. is remarkable discovery of fuzzy set theory paved a different way for dealing with uncertainties in various domains of science and technology.
Graph theory is moving quickly into the mainstream of mathematics, primarily due to its applications in engineering, communication networks, computer science, and artificial intelligence. In 1973, Kauffmann [2] introduced the notion of fuzzy graph, which is based on Zadeh's fuzzy relation [3]. Another elaborated definition of fuzzy graph was introduced by Rosenfeld [4]. Bhattacharya [5] subsequently gave some helpful results on fuzzy graphs and some operations on fuzzy graph theory were explored by Mordeson and Nair [6]. Many researchers studied fuzzy graphs in recent decades [7][8][9].
However, the theory of fuzzy sets has inadequacy to deal with parametrization tool. Soft set theory proposed by Molodtsov [10] has the ability to cope with this difficulty and is defined as a pair (ξ, M), where ξ is a mapping given by ξ: M ⟶ P(E). Soft sets have been generalized to numerous directions beginning with Maji et al. [11,12] who introduced fuzzy soft sets and Ahmad and Kharal [13] discussed some properties of fuzzy soft sets. In algebraic structures, soft sets and their hybrid models based on fuzzy soft sets, generalized fuzzy soft sets, rough soft sets, and soft rough sets have been implemented effectively [14][15][16][17][18][19]. Sarwar [20] elaborated the notion of rough graph and discussed decision-making approaches based on rough numbers and rough graphs. Akram and Nawaz [21] introduced the concepts of fuzzy soft graphs (named as fuzzy type-1 soft graph), vertex-induced soft graphs, and edge-induced soft graphs and also discussed some operations on soft graphs. Akram and Zafar [22] introduced various hybrid models based on fuzzy sets, soft sets, and rough sets. Further, Akram in cooperation with other researchers [23][24][25][26] discussed various applications and extensions of graph theory to study different types of uncertainties in real-world problems. Nowadays, researchers are actively working on interval type-2 fuzzy arc lengths [27], trapezoidal interval type-2 fuzzy soft sets [28], total uniformity of graph under fuzzy soft information [29], fuzzy soft cycles [30], and fuzzy soft β−coverings.
All these existing models have the same restriction that one cannot freely select the parameters. at is, if a correspondence or association occurs between parameters, then none of these models can solve the problems completely. Chatterjee et al. [31] proposed the concept of type-2 soft sets to deal with the correspondence between parameters, which is a generalization of Molodtsov's soft sets (called type-1 soft sets). Type-2 soft sets reparameterize the already parameterized crisp sets and thus have more freedom and effectiveness in dealing with imprecision as compared to type-1 soft sets. Hayat et al. [32][33][34] introduced vertex-neighborsbased type-2 soft sets, type-2 soft graphs, and irregular type-2 soft graphs and presented certain types of type-2 soft graphs. e motives of this study are as follows: (1) Soft sets and their hybrid models are used to deal with uncertainty based on parametrization tool. e correspondence, association, or relation occurring among parameters cannot be discussed with existing approaches. Type-2 soft models tackle this difficulty and present a mathematical approach to reparameterize the existing soft models. To deal with partial membership of objects, the main focus of this study is to introduce a hybrid model by combining fuzzy set theory with type-2 soft sets. (2) Graph theory is an essential approach to study relations among objects using a figure consisting of vertices and lines joining these vertices. But there is an information loss in graphical models whether the objects are fully related or partially related, that is, uncertain and parameterized relations among objects. To handle this information loss, there is a need to represent the graphical models under fuzzy type-2 soft environment.
e main contribution of this study is as follows: (1) e present study introduces the mathematical approaches of vertex-neighbors-based type-2 soft set and vertex-neighbors-based type-2 soft graphs under fuzzy environment. e notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles are discussed with certain operations and numerical examples. (2) e importance of presented concepts is studied with an application in chemical digestion and national engineering services.

Preliminaries
e term crisp graph on a nonvoid universe (named as set of vertices) J is defined as a pair G � (J, K), where K ⊆ J × J is named as set of edges. Crisp graph (J, K) is a special case of the fuzzy graph with each vertex and edge of (J, K) having degree of membership 1. A soft graph corresponding to a crisp graph G is a parameterized family of subgraphs of G. A soft graph on a nonempty set J is a 3−tuple (J, K, A) such that, for each e ∈ A, (J(e), K(e)) is a graph, where J(e) ⊆ J and K(e) ⊆ J(e) × J(e).
Definition 1 (see [31]). Let (E, M) be a soft universe and let η(E) be the set of all T1SSs over (E, M). en a mapping W: We refer to the parameter set B as the "primary set of parameters" although the collection of parameters denoted by ∪ F (δ) is called "underlying set of parameters." Definition 2 (see [32]). Suppose that G � (J, K) is a simple graph. Suppose that B ⊂ J and Γ(J) is the set of all T1SSs over J. Suppose that [ξ * , B] is a T2SS over J. en a mapping ξ * : B ⟶ Γ(J) is said to be a T2SS over J and is denoted as Definition 3 (see [32]). Suppose that G � (J, K) is a simple graph. Suppose that B ⊂ J and Γ(K) is the set of all T1SSs over K. Suppose that [ξ * , B] is a VN-T2SS over J. en a mapping ψ * : B ⟶ Γ(K) is said to be a T2SS over K and is denoted as We present the notations that are used in this research article in Table 1. FT1SS. us, for each e ∈ B, there exists a FT1SS (S e , L e ) such that S * (e) � (S e , L e ), where S e : L e ⟶ P(E) and L e ⊂ M. In this case, we refer to the parameter set B as the "primary set of parameters," while the set of parameters ∪ L e is known as the "underlying set of parameters." Definition 5. Let G � (J, K) be a fuzzy graph. e set of neighbors of an element (j, μ(j)) is denoted by NB j and defined by can be explained as is FT2SS is said to be a vertex-neighbors fuzzy type-2 soft set (VN-FT2SS) over J.
Definition 7. Let G � (J, K) be a fuzzy graph. Suppose that B ⊂ J and Γ(K) is the set of all T1SSs over K. Suppose en a mapping ψ: B ⟶ Γ(K) is said to be a FT2SS over K and is denoted as [ψ, B]. For every vertex j ∈ B, ψ(j) � (ψ (j) , NB j ) is a FT1SS and ψ (j) : NB j ⟶ P(K) can be explained as respectively. If (ξ j (u), ψ j (u)) ∀u ∈ NB j represent a fuzzy graph in fuzzy type-2 soft graph G, then (ξ(j), ψ(j)) ∀j ∈ B is called FT1SG.
is called a fuzzy type-2 soft graph (FT2SG) if it satisfies the following conditions: (1) Fuzzy type-2 soft graph G � (Z(e 3 )) is shown in Figure 2.
Definition 9. Let G � (G, ξ, ψ, B, NB B ) be a fuzzy type-2 soft graph; the complement of G is denoted by G c and defined by G c � (Z c (z 1 ), Z c (z 2 ),... , Z c (z n )) for all Example 2. Let G � (J, K) be a fuzzy graph as shown in Figure 3.
Proof. Let G be a regular FT2SG. Suppose that (Z σ , NB σ ) is a FT1SG corresponding to Z(σ) for all σ ∈ B; then Z σ (j) for all j ∈ NB σ must be a regular fuzzy graph. As we know that complement of a regular graph is regular, Z c σ (j) ∀j ∈ NB σ is also a regular fuzzy graph. It provides FT1SG corresponding to a Z c (σ) for all σ ∈ B being regular FT1SG. us, G c is a regular FT2SG of G. □ Definition 11. Let G be a FT2SG; G is said to be an irregular FT2SG if FT1SG corresponding to Z(χ) is an irregular FT1SG for all χ ∈ B. Let [ξ, B] and [ψ, B] be two FT2SSs over J and K, respectively. We have Define en G � (Z(e 4 ), Z(e 5 )) is an irregular FT2SG as shown in Figure 6.  Let [ξ, B] and [ψ, B] be two FT2SSs over J and K, respectively. We have Define en G � (Z(e 3 ), Z(e 5 )) is a neighborly irregular FT2SG as shown in Figure 8.
Example 5. Let G � (J, K) be a fuzzy graph as shown in Figure 9.     10. It can also be defined as VN-type-2 soft tree.
On the contrary, assume that G is a complete FT2SG; then each Z χ (j) ∀j ∈ NB χ will also be complete.
Let v, w be arbitrary nodes of Z χ (j) joined by a line vw. Since Z χ (j) having n ≥ 3 vertices of G is a FT1SG, then a minimum one vertex η which is connected to v by an edge vη and to w by an edge wη as Z χ (j) be a complete fuzzy graph. en there is a cycle vwηv. erefore, Z χ (j) ∀j ∈ NB χ cannot be a FT1ST, which is opposite to the fact that Z χ (j) is a connected FT1SG of FT2SG. So, G is not a complete FT2SG.
Example 6. Let G � (J, K) be a fuzzy graph as shown in Figure 11, where Define We can check that G � (Z(e 9 ), Z(e 8 )) is a FT2SC as shown in Figure 12. It is also defined as a fuzzy VN-type-2 soft cycle.
Example 7. Let G � (J, K) be a fuzzy graph as shown in Figure 13.
and [ψ, B] be two FT2SSs over J and K, respectively. We have

Journal of Mathematics 9
Define Z(a) � (ξ(a), ψ(a)) and Z(b) � (ξ(b), ψ(b)) are FT1SGs as shown in Figure 14. We can see that Proof. Let G be a FT2SC. Let (Z χ , NB χ ) be a T1FSC corresponding to Z(χ) for every χ ∈ B. en, Z χ (j) is a cycle ∀j ∈ NB χ . We know that cycle is a path that is closed and every vertex of cycle is of degree 2; this signifies that Z χ (j) is a regular fuzzy graph for all j ∈ NB χ . erefore, Define en Z a (d) Figure 14: Z(a)).
Proof. Let G 2 be a FT2SST of G 1 . en, by using the definition of FT2SST, Since FT1ST corresponding to Z 2 (j) is a FT1SST of FT1ST corresponding to Z 1 (j) for all j ∈ B 2 , we have ξ 2 ⊆ ξ 1 and ψ 2 ⊆ ψ 1 ∀j ∈ B 2 . Conversely, we have ξ 2 (j) ⊆ ξ 1 (j) and ψ 2 (j) ⊆ ψ 1 (j) ∀j ∈ B 2 . As G 1 is a fuzzy type-2 soft tree, fuzzy type-1 soft set corresponding to Z 1 (j) forms a FT1ST of G 2 for all j ∈ B 1 . Also, G 2 is a fuzzy type-2 soft tree, and fuzzy type-1 soft set corresponding to Z 2 (j) forms a FT1ST of G 1 for all j ∈ B 2 .
It can be written as Example 9. Let G be a fuzzy graph as shown in Figure 18

Applications of Fuzzy Type-2 Soft Graphs
In this section, we apply the concept of fuzzy type-2 soft graphs to decision-making problems in chemical digestion and national engineering services. e selection of a suitable object problem can be considered as a decisionmaking problem, in which final identification of object is decided on a given set of information. A detailed description of the algorithm for the selection of most suitable object based on available set of parameters is given in Algorithm 1 and the flow chart shown in Figure 23; purposed algorithm can be used to find out the best correspondence relationship between the neighboring objects in the decision-making problem. is method can be applied in various domains for multicriteria selection of objects.

Determination of Dominant Food Components in
Chemical Digestion. We present an application of FT2SG in chemical digestion and discuss how to apply FT2SG in chemical digestion of spinach. Spinach is generally composed of carbohydrates, protein, lipids, minerals, vitamins, and nucleic acids. We mainly focused on the digestion of carbohydrates, proteins, lipids, and nucleic acids, which is carried out by a variety of salivary enzymes and the enzymes present in other parts of digestive system; that is, amylase, pepsin, and trypsin are released as a result of involuntary signal generated by our body to digest the food. When 25 g of spinach is taken, it contains carbohydrates (0.9g), protein (0.7g), lipids (0.1g), nucleic acids (0.3g), involuntary signal (0.3), pepsin (0.2), amylase (0.2), and trypsin (0.1), represented as vertices donated by e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 , e 8 , respectively. "Chemical digestion" is the enzyme-mediated, hydrolysis method that converts large macronutrients into smaller molecules.
(i) Carbohydrate mostly comprises amylose and glycogen. Long carbohydrates chains are broken down into disaccharides which are decomposed by amylase enzyme.
(ii) Proteins are usually broken down into amino acids by peptidase enzyme as well as trypsin and chymotrypsin. (iii) Lipids are hydrolyzed by pancreatic lipase enzyme. (iv) Nucleic acids, that is, DNA and RNA, are hydrolyzed by pancreatic nuclease. (v) Involuntary signal is generated by the brain in order to carry out chemical digestion in the digestive system.
Protein digestion occurs in stomach and duodenum by the action of three primary enzymes.
(i) Pepsin, disguised by abdomen (ii) Trypsin, disguised through pancreas (iii) Amylase, disguised through saliva and pancreas Note that the values of pepsin, trypsin, amylase, and involuntary signal are supposed as we cannot calculate the amounts of these products released as a result of con-

) Construct the resultant VN-fuzzy graph by taking the intersection of vertex-neighbors fuzzy graphs
In fuzzy graph (J, K) as shown in Figure 24, edges represent the amount of energy utilized by the body in order to carry out the digestion process. Let B � (e 1 , 0.9), (e 2 , 0.7) represent the amounts of carbohydrates and protein released when 25 g of spinach is consumed. We have NB e 1 � (e 2 , 0.7), (e 4 , 0.3) , NB e 2 � (e 3 , 0.1), (e 1 , 0.9) .
Let [ξ, B] and [ψ, B] be two FT2SSs over J and K, respectively. We have

Water Supply for National Engineering Services.
We present the application of fuzzy type-2 soft graph in the National Engineering Services Pakistan (NESPAK). e National Engineering Services Pakistan is a Pakistani multinational state-owned corporation that provides construction, management, and consulting services globally. Every government project has something to do with NES-PAK at some time of its planning or implementation. In fuzzy graph G � (J, K) as shown in Figure 27, vertices represent some important projects.
FT1SGs corresponding to Z(e) � (ξ(e), ψ(e)) and Z(d) � (ξ(d), ψ(d)), respectively, are shown in fuzzy type-2 soft graph 28. (Figure 28) e tabular representations of resultant vertex-neighbors fuzzy graphs Z * (e) and Z * (d) shown in Figure 29 with the choice values C j i � k S ik for all i, k are given in Tables 4 and 5. e decision value is S i � ∨ 6 i ( ∧ j C j i ) � ∨ 6 i�1 0.7 ∧ 0.7, { 0.6 ∧ 1.1, 0 ∧ 0.4, 0.6 ∧ 0.6, 0.6 ∧ 0.5, 0.1 ∧ 0.1} � 0.7, from the choice value C j i of fuzzy type-2 soft graphs for j � 1, 2. e optimal project is "a � water supply." So, NESPAK provides the best engineering services to the project of "water supply." Advantages of the Proposed Method. e advantages of the proposed method based on FT2SGs are as follows: (1) e method can be effectively used to handle uncertainty and vagueness with correspondence, assertion, and relations among parameters.
(2) e proposed method incorporates parametrization tool with fuzzy information to effectively handle more uncertain conditions and errors in given data.
(3) e presented method considers vertex-neighbors coordination tool along with reparametrization to study the interrelationship and ambiguity among objects.

Comparison Analysis
In this section, we discuss the comparison of fuzzy type-2 soft graphs with fuzzy soft graphs and type-2 soft graphs.

Comparison with Fuzzy Soft
Graphs. Fuzzy soft graph [21] is a parameterized family of fuzzy graphs, and it is an extension of a soft graph. e fuzzy type-2 soft graph is a parameterized family of VN-fuzzy soft graphs and an extension of type-2 soft graph. Fuzzy type-2 soft graphs show    Tables 6 and 7. e fuzzy graphs H(e 1 ) and H(e 2 ) of fuzzy soft graph G � H(e 1 ), H(e 2 ) corresponding to the parameters "carbohydrates" and "protein" are shown in Figure 30.
e fuzzy graphs H(e 1 ) and H(e 2 ) and the choice values C k i � j S ij for all i, j, k � 1, 2 are given in Tables 8 and 9, respectively. e decision value is Z(d) Figure 28: Fuzzy type-2 soft graph for national engineering services.

22
Journal of Mathematics suitable object determined by fuzzy soft graph as above and fuzzy type-2 soft graph in Section 4.1 is dependent on information determined by selected set of parameters and fuzzy values in VN-fuzzy graphs, respectively. As the coordination among objects varies, the solution changes accordingly. So, in this case, when the objects show close vertex-neighbors coordination according to observed data, fuzzy type-2 soft graph model can be used and in the case when fuzzy relations are given along with different parameters, fuzzy soft graph model can be used.

Comparison with Type-2 Soft Graphs.
In structure of a graph, the vertex-neighbors correspondence has an important role. e type-2 soft graph [32] is based on the correspondence of initial parameters (vertex soft set) and underlying parameters. e type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors' structure and is applicable in computational analysis, applied intelligence, and decision-making problems. e theory of fuzzy sets has played an important role to form useful models for handling partial membership of objects. To overcome the parameterized limitations of fuzzy set, the theory of fuzzy type-2 soft set was introduced. Fuzzy type-2 soft graph model is a more efficient model as compared to type-2 soft graph model to represent the parametric uncertainty in graphical networks. It is observed that, for the selection of dominant food components in chemical digestion using given type-2 soft information, we are not able to identify any object (dominating component). In this case, the simple type-2 soft information provides no solution.
To determine the solution of the problem, it is necessary to have fuzzy information or define a fuzzy relation in order to attain a suitable approximation approach for selecting at least one object. So, fuzzy type-2 soft graph is more reliable in such decision-making problems.

Conclusions and Future Directions
Molodtsov's soft set theory is an effective and rational approach to understand uncertainties in terms of parameters. Type-2 soft sets have been introduced by adding the primary relations among parameters in soft sets. We have introduced the notions of fuzzy type-2 soft sets and fuzzy type-2 soft graphs to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. We have discussed certain properties of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We have discussed different methods of construction of fuzzy type-2 soft graphs with certain operations and elaborated these concepts with numerical examples. We have studied the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services. e present study can be extended to various directions including (1) Pythagorean fuzzy type-2 soft graphs, (2) spherical fuzzy type-2 soft graphs, and (3) picture fuzzy type-2 soft trees.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.